Solved: A particle on a spring moves in simple harmonic

The distance between the crest of a water wave and the next trough is 2 m. If the frequency of a particular wave is 2 Hz, what is the speed of the wave? (a) 4 m/s (b) 1 m/s (c) 8 m/s (d) 2 m/s (e) impossible to determine from the information given

The position of an object moving with simple harmonic motion is given by x 5 4 cos (6pt), where x is in meters and t is in seconds. What is the period of the oscillating system? (a) 4 s (b) 16 s (c) 13 s (d) 6p s (e) impossible to determine from the information given

A block-spring system vibrating on a frictionless, horizontal surface with an amplitude of 6.0 cm has a total energy of 12 J. If the block is replaced by one having twice the mass of the original block and the amplitude of the motion is again 6.0 cm, what is the energy of the more massive system? (a) 12 J (b) 24 J (c) 6 J (d) 48 J (e) none of those answers

A mass of 0.40 kg, hanging from a spring with a spring constant of 80.0 N/m, is set into an up-and-down simple harmonic motion. If the mass is displaced from equilibrium by 0.10 m and released from rest, what is its speed when moving through the equilibrium point? (a) 0 (b) 1.4 m/s (c) 2.0 m/s (d) 3.4 m/s (e) 4.2 m/s

If an object of mass m attached to a light spring is replaced by one of mass 9m, the frequency of the vibrating system changes by what multiplicative factor? (a) 19 (b) 13 (c) 3.0 (d) 9.0 (e) 6.0

An object of mass 0.40 kg, hanging from a spring with a spring constant of 8.0 N/m, is set into an up-and-down simple harmonic motion. What is the magnitude of the acceleration of the object when it is at its maximum displacement of 0.10 m? (a) 0 (b) 0.45 m/s2 (c) 1.0 m/s2 (d) 2.0 m/s2 (e) 2.40 m/s2

A runaway railroad car with mass 3.0 3 105 kg coasts across a level track at 2.0 m/s when it collides elastically with a spring-loaded bumper at the end of the track. If the spring constant of the bumper is 2.0 3 106 N/m, what is the maximum compression of the spring during the collision? (a) 0.77 m (b) 0.58 m (c) 0.34 m (d) 1.07 m (e) 1.24 m

If a simple pendulum oscillates with a small amplitude and its length is doubled, what happens to the frequency of its motion? (a) It doubles. (b) It becomes !2 times as large. (c) It becomes half as large. (d) It becomes 1/!2 as large. (e) It remains the same.

A simple pendulum has a period of 2.5 s. What is its period if its length is made four times as large? (a) 0.625 s (b) 1.25 s (c) 2.5 s (d) 3.54 s (e) 5.0 s

A particle on a spring moves in simple harmonic motion along the x-axis between turning points at x1 5 100 cm and x2 5 140 cm. At which of the following positions does the particle have its maximum kinetic energy? (a) 100 cm (b) 110 cm (c) 120 cm (d) 130 cm (e) 140 cm

Which of the following statements is not true regarding a massspring system that moves with simple harmonic motion in the absence of friction? (a) The total energy of the system remains constant. (b) The energy of the system is continually transformed between kinetic and potential energy. (c) The total energy of the system is proportional to the square of the amplitude. (d) The potential energy stored in the system is greatest when the mass passes through the equilibrium position. (e) The velocity of the oscillating mass has its maxi- mum value when the mass passes through the equilibrium position.

A block is attached to a spring hanging vertically. After being slowly lowered, it hangs at rest with the spring stretched by 15.0 cm. If the block is raised back up and released from rest with the spring unstretched, what maximum distance does it fall? (a) 7.5 cm (b) 15.0 cm (c) 30.0 cm (d) 60.0 cm (e) impossible to determine without knowing the mass and spring constant

An objectspring system moving with simple harmonic motion has an amplitude A. When the kinetic energy of the object equals twice the potential energy stored in the spring, what is the position x of the object? (a) A (b) 13 A (c) A/!3 (d) 0 (e) none of those answers

You stand on the end of a diving board and bounce to set it into oscillation. You find a maximum response in terms of the amplitude of oscillation of the end of the board when you bounce at a frequency f. You now move to the middle of the board and repeat the experiment. Is the resonance frequency of the forced oscillations at this point (a) higher, (b) lower, or (c) the same as f ?

A horizontal blockspring system with the block on a frictionless surface has total mechanical energy E 5 47.0 J and a maximum displacement from equilibrium of 0.240 m. (a) What is the spring constant? (b) What is the kinetic energy of the system at the equilibrium point? (c) If the maximum speed of the block is 3.45 m/s, what is its mass? (d) What is the speed of the block when its displacement is 0.160 m? (e) Find the kinetic energy of the block at x 5 0.160 m. (f) Find the potential energy stored in the spring when x 5 0.160 m. (g) Suppose the same system is released from rest at x 5 0.240 m on a rough surface so that it loses 14.0 J by the time it reaches its first turning point (after passing equilibrium at x 5 0). What is its position at that instant?

A 0.250-kg block resting on a frictionless, horizontal surface is attached to a spring having force constant 83.8 N/m as in Figure P13.16. A horizontal force F S causes the spring to stretch a distance of 5.46 cm from its equilibrium position. (a) Find the value of F. (b) What is the total energy stored in the system when the spring is stretched? (c) Find the magnitude of the acceleration of the block immediately after the applied force is removed. (d) Find the speed of the block when it first reaches the equilibrium position. (e) If the surface is not frictionless but the block still reaches the equilibrium position, how would your answer to part (d) change? (f) What other information would you need to know to find the actual answer to part (d) in this case?

A 0.40-kg object connected to a light spring with a force constant of 19.6 N/m oscillates on a frictionless horizontal surface. If the spring is compressed 4.0 cm and released from rest, determine (a) the maximum speed of the object, (b) the speed of the object when the spring is compressed 1.5 cm, and (c) the speed of the object as it passes the point 1.5 cm from the equilibrium position. (d) For what value of x does the speed equal one-half the maximum speed?

An objectspring system oscillates with an amplitude of 3.5 cm. If the spring constant is 250 N/m and the object has a mass of 0.50 kg, determine (a) the mechanical energy of the system, (b) the maximum speed of the object, and (c) the maximum acceleration of the object.

At an outdoor market, a bunch of bananas attached to the bottom of a vertical spring of force constant 16.0 N/m is set into oscillatory motion with an amplitude of 20.0 cm. It is observed that the maximum speed of the bunch of bananas is 40.0 cm/s. What is the weight of the bananas in newtons?

A 50.0-g object is attached to a horizontal spring with a force constant of 10.0 N/m and released from rest with an amplitude of 25.0 cm. What is the velocity of the object when it is halfway to the equilibrium position if the surface is frictionless?

A horizontal spring attached to a wall has a force constant of k 5 850 N/m. A block of mass m 5 1.00 kg is attached to the spring and rests on a frictionless, horizontal surface as in Figure P13.21 on page 468. (a) The block is pulled to a position xi 5 6.00 cm from equilibrium and released. Find the potential energy stored in the spring when the block is 6.00 cm from equilibrium. (b) Find the speed of the block as it passes through the equilibrium position. (c) What is the speed of the block when it is at a position xi/2 5 3.00 cm?

An object moves uniformly around a circular path of radius 20.0 cm, making one complete revolution every 2.00 s. What are (a) the translational speed of the object, (b) the frequency of motion in hertz, and (c) the angular speed of the object?

Consider the simplified single-piston engine in Figure P13.23. If the wheel rotates at a constant angular speed v, explain why the piston oscillates in simple harmonic motion.

The period of motion of an objectspring system is T 5 0.528 s when an object of mass m 5 238 g is attached to the spring. Find (a) the frequency of motion in hertz and (b) the force constant of the spring. (c) If the total energy of the oscillating motion is 0.234 J, find the amplitude of the oscillations.

A vertical spring stretches 3.9 cm when a 10-g object is hung from it. The object is replaced with a block of mass 25 g that oscillates up and down in simple harmonic motion. Calculate the period of motion.

When four people with a combined mass of 320 kg sit down in a 2.0 3 103-kg car, they find that their weight compresses the springs an additional 0.80 cm. (a) What is the effective force constant of the springs? (b) The four people get out of the car and bounce it up and down. What is the frequency of the cars vibration?

A cart of mass 250 g is placed on a frictionless horizontal air track. A spring having a spring constan of 9.5 N/m is attached between the cart and the left end of the track. If the cart is displaced 4.5 cm from its equilibrium position, find (a) the period at which it oscillates, (b) its maximum speed, and (c) its speed when it is located 2.0 cm from its equilibrium position.

The position of an object connected to a spring varies with time according to the expression x 5 (5.2 cm) sin (8.0pt). Find (a) the period of this motion, (b) the frequency of the motion, (c) the amplitude of the motion, and (d) the first time after t 5 0 that the object reaches the position x 5 2.6 cm.

A 326-g object is attached to a spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is 5.83 J, find (a) the maximum speed of the object, (b) the force constant of the spring, and (c) the amplitude of the motion.

An object executes simple harmonic motion with an amplitude A. (a) At what values of its position does its speed equal half its maximum speed? (b) At what values of its position does its potential energy equal half the total energy?

A 2.00-kg object on a frictionless horizontal track is attached to the end of a horizontal spring whose force constant is 5.00 N/m. The object is displaced 3.00 m to the right from its equilibrium position and then released, initiating simple harmonic motion. (a) What is the force (magnitude and direction) acting on the object 3.50 s after it is released? (b) How many times does the object oscillate in 3.50 s?

A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 7.50 N is applied. A 0.500-kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x 5 5.00 cm and released from rest at t 5 0. (a) What is the force constant of the spring? (b) What are the angular frequency v, the frequency, and the period of the motion? (c) What is the total energy of the system? (d) What is the amplitude of the motion? (e) What are the maximum velocity and the maximum acceleration of the particle? (f) Determine the displacement x of the particle from the equilibrium position at t 5 0.500 s. (g) Determine the velocity and acceleration of the particle when t 5 0.500 s.

Given that x 5 A cos (vt) is a sinusoidal function of time, show that v (velocity) and a (acceleration) are also sinusoidal functions of time. Hint: Use Equations 13.6 and 13.2.

A man enters a tall tower, needing to know its height. He notes that a long pendulum extends from the ceiling almost to the floor and that its period is 15.5 s. (a) How tall is the tower? (b) If this pendulum is taken to the Moon, where the free-fall acceleration is 1.67 m/s2, what is the period there?

A simple pendulum has a length of 52.0 cm and makes 82.0 complete oscillations in 2.00 min. Find (a) the period of the pendulum and (b) the value of g at the location of the pendulum.

A seconds pendulum is one that moves through its equilibrium position once each second. (The period of the pendulum is 2.000 s.) The length of a seconds pendulum is 0.992 7 m at Tokyo and 0.994 2 m at Cambridge, England. What is the ratio of the free-fall accelerations at these two locations?

A pendulum clock that works perfectly on the Earth is taken to the Moon. (a) Does it run fast or slow there? (b) If the clock is started at 12:00 midnight, what will it read after one Earth day (24.0 h)? Assume the free-fall acceleration on the Moon is 1.63 m/s2.

A coat hanger of mass m 5 0.238 kg oscillates on a peg as a physical pendulum as shown in Figure P13.38. The distance from the pivot to the center of mass of the coat hanger is d 5 18.0 cm and the period of the motion is T 5 1.25 s. Find the moment of inertia of the coat hanger about the pivot.

The free-fall acceleration on Mars is 3.7 m/s2. (a) What length of pendulum has a period of 1 s on Earth? (b) What length of pendulum would have a 1-s period on Mars? An object is suspended from a spring with force constant 10 N/m. Find the mass suspended from this spring that would result in a period of 1 s (c) on Earth and (d) on Mars.

A simple pendulum is 5.00 m long. (a) What is the period of simple harmonic motion for this pendulum if it is located in an elevator accelerating upward at 5.00 m/s2? (b) What is its period if the elevator is accelerating downward at 5.00 m/s2? (c) What is the period of simple harmonic motion for the pendulum if it is placed in a truck that is accelerating horizontally at 5.00 m/s2?

The sinusoidal wave shown in Figure P13.41 is traveling in the positive x-direction and has a frequency of 18.0 Hz. Find the (a) amplitude, (b) wavelength, (c) period, and (d) speed of the wave.

An object attached to a spring vibrates with simple harmonic motion as described by Figure P13.42. For this motion, find (a) the amplitude, (b) the period, (c) the angular frequency, (d) the maximum speed, (e) the maximum acceleration, and (f) an equation for its position x in terms of a sine function.

Light waves are electromagnetic waves that travel at 3.00 3 108 m/s. The eye is most sensitive to light having a wavelength of 5.50 3 1027 m. Find (a) the frequency of this light wave and (b) its period.

The distance between two successive minima of a transverse wave is 2.76 m. Five crests of the wave pass a given point along the direction of travel every 14.0 s. Find (a) the frequency of the wave and (b) the wave speed.

A harmonic wave is traveling along a rope. It is observed that the oscillator that generates the wave completes 40.0 vibrations in 30.0 s. Also, a given maximum travels 425 cm along the rope in 10.0 s. What is the wavelength?

A bat can detect small objects, such as an insect, whose size is approximately equal to one wavelength of the sound the bat makes. If bats emit a chirp at a frequency of 60.0 3 103 Hz and the speed of sound in air is 343 m/s, what is the smallest insect a bat can detect?

A cork on the surface of a pond bobs up and down two times per second on ripples having a wavelength of 8.50 cm. If the cork is 10.0 m from shore, how long does it take a ripple passing the cork to reach the shore?

Ocean waves are traveling to the east at 4.0 m/s with a distance of 20 m between crests. With what frequency do the waves hit the front of a boat (a) when the boat is at anchor and (b) when the boat is moving westward at 1.0 m/s?

An ethernet cable is 4.00 m long and has a mass of 0.200 kg. A transverse wave pulse is produced by plucking one end of the taut cable. The pulse makes four trips down and back along the cable in 0.800 s. What is the tension in the cable?

A circus performer stretches a tightrope between two towers. He strikes one end of the rope and sends a wave along it toward the other tower. He notes that it takes the wave 0.800 s to reach the opposite tower, 20.0 m away. If a 1.00-m length of the rope has a mass of 0.350 kg, find the tension in the tightrope.

A piano string of mass per unit length 5.00 3 103 kg/m is under a tension of 1 350 N. Find the speed with which a wave travels on this string.

A student taking a quiz finds on a reference sheet the two equations f 5 1 T and v 5 T m She has forgotten what T represents in each equation. (a) Use dimensional analysis to determine the units required for T in each equation. (b) Explain how you can identify the physical quantity each T represents from the units.

Transverse waves with a speed of 50.0 m/s are to be produced on a stretched string. A 5.00-m length of string with a total mass of 0.060 0 kg is used. (a) What is the required tension in the string? (b) Calculate the wave speed in the string if the tension is 8.00 N.

An astronaut on the Moon wishes to measure the local value of g by timing pulses traveling down a wire that has a large object suspended from it. Assume a wire of mass 4.00 g is 1.60 m long and has a 3.00-kg object suspended from it. A pulse requires 36.1 ms to traverse the length of the wire. Calculate gMoon from these data. (You may neglect the mass of the wire when calculating the tension in it.)

A simple pendulum consists of a ball of mass 5.00 kg hanging from a uniform string of mass 0.060 0 kg and length L. If the period of oscillation of the pendulum is 2.00 s, determine the speed of a transverse wave in the string when the pendulum hangs vertically.

A string is 50.0 cm long and has a mass of 3.00 g. A wave travels at 5.00 m/s along this string. A second string has the same length, but half the mass of the first. If the two strings are under the same tension, what is the speed of a wave along the second string?

Tension is maintained in a string as in Figure P13.57. The observed wave speed is v 5 24.0 m/s when the suspended mass is m 5 3.00 kg. (a) What is the mass per unit length of the string? (b) What is the wave speed when the suspended mass is m 5 2.00 kg?

The elastic limit of a piece of steel wire is 2.70 3 109 Pa. What is the maximum speed at which transverse wave pulses can propagate along the wire without exceeding its elastic limit? (The density of steel is 7.86 3 103 kg/m3.)

A 2.65-kg power line running between two towers has a length of 38.0 m and is under a tension of 12.5 N. (a) What is the speed of a transverse pulse set up on the line? (b) If the tension in the line was unknown, describe a procedure a worker on the ground might use to estimate the tension.

A taut clothesline has length L and a mass M. A transverse pulse is produced by plucking one end of the clothesline. If the pulse makes n round trips along the clothesline in t seconds, find expressions for (a) the speed of the pulse in terms of n, L, and t and (b) the tension F in the clothesline in terms of the same variables and mass M.

A wave of amplitude 0.30 m interferes with a second wave of amplitude 0.20 m traveling in the same direction. What are (a) the largest and (b) the smallest resultant amplitudes that can occur, and under what conditions will these maxima and minima arise?

The position of a 0.30-kg object attached to a spring is described by x 5 (0.25 m) cos (0.4pt) Find (a) the amplitude of the motion, (b) the spring constant, (c) the position of the object at t 5 0.30 s, and (d) the objects speed at t 5 0.30 s.

An object of mass 2.00 kg is oscillating freely on a vertical spring with a period of 0.600 s. Another object of unknown mass on the same spring oscillates with a period of 1.05 s. Find (a) the spring constant k and (b) the unknown mass.

A certain tuning fork vibrates at a frequency of 196 Hz while each tip of its two prongs has an amplitude of 0.850 mm. (a) What is the period of this motion? (b) Find the wavelength of the sound produced by the vibrating fork, taking the speed of sound in air to be 343 m/s.

A simple pendulum has mass 1.20 kg and length 0.700 m. (a) What is the period of the pendulum near the surface of Earth? (b) If the same mass is attached to a spring, what spring constant would result in the period of motion found in part (a)?

A 500-g block is released from rest and slides down a frictionless track that begins 2.00 m above the horizontal, as shown in Figure P13.66. At the bottom of the track, where the surface is horizontal, the block strikes and sticks to a light spring with a spring constant of 20.0 N/m. Find the maximum distance the spring is compressed.

A 3.00-kg object is fastened to a light spring, with the intervening cord passing over a pulley (Fig. P13.67). The pulley is frictionless, and its inertia may be neglected. The object is released from rest when the spring is unstretched. If the object drops 10.0 cm before stopping, find (a) the spring constant of the spring and (b) the speed of the object when it is 5.00 cm below its starting point.

A 5.00-g bullet moving with an initial speed of 400 m/s is fired into and passes through a 1.00-kg block, as in Figure P13.68. The block, initially at rest on a frictionless horizontal surface, is connected to a spring with a spring constant of 900 N/m. If the block moves 5.00 cm to the right after impact, find (a) the speed at which the bullet emerges from the block and (b) the mechanical energy lost in the collision.

A large block P executes horizontal simple harmonic motion as it slides across a frictionless surface with a frequency f 5 1.50 Hz. Block B rests on it, as shown in Figure P13.69, and the coefficient of static friction between the two is ms 5 0.600. What maximum amplitude of oscillation can the system have if block B is not to slip?

A spring in a toy gun has a spring constant of 9.80 N/m and can be compressed 20.0 cm beyond the equilibrium position. A 1.00-g pellet resting against the spring is propelled forward when the spring is released. (a) Find the muzzle speed of the pellet. (b) If the pellet is fired horizontally from a height of 1.00 m above the floor, what is its range?

A light balloon filled with helium of density 0.179 kg/m3 is tied to a light string of length L 5 3.00 m. The string is tied to the ground, forming an inverted simple pendulum (Fig. P13.71a). If the balloon is displaced slightly from equilibrium, as in Figure P13.71b, (a) show that the motion is simple harmonic and (b) determine the period of the motion. Take the density of air to be 1.29 kg/m3. Hint: Use an analogy with the simple pendulum discussed in the text, and see Chapter 9.

An object of mass m is connected to two rubber bands of length L, each under tension F, as in Figure P13.72. The object is displaced vertically by a small distance y. Assuming the tension does not change, show that (a) the restoring force is 2(2F/L)y and (b) the system exhibits simple harmonic motion with an angular frequency v 5 !2F/mL.

Assume a hole is drilled through the center of the Earth. It can be shown that an object of mass m at a distance r from the center of the Earth is pulled toward the center only by the material in the shaded portion of Figure P13.73. Assume Earth has a uniform density r. Write down Newtons law of gravitation for an object at a distance r from the center of the Earth and show that the force on it is of the form of Hookes law, F 5 2kr, with an effective force constant of k 5 143 2prGm, where G is the gravitational constant.

Figure P13.74 shows a crude model of an insect wing. The mass m represents the entire mass of the wing, which pivots about the fulcrum F. The spring represents the surrounding connective tissue. Motion of the wing corresponds to vibration of the spring. Suppose the mass of the wing is 0.30 g and the effective spring constant of the tissue is 4.7 3 1024 N/m. If the mass m moves up and down a distance of 2.0 mm from its position of equilibrium, what is the maximum speed of the outer tip of the wing?

A 2.00-kg block hangs without vibrating at the end of a spring (k 5 500 N/m) that is attached to the ceiling of an elevator car. The car is rising with an upward acceleration of g/3 when the acceleration suddenly ceases (at t 5 0). (a) What is the angular frequency of oscillation of the block after the acceleration ceases? (b) By what amount is the spring stretched during the time that the elevator car is accelerating?

A system consists of a vertical spring with force constant k 5 1 250 N/m, length L 5 1.50 m, and object of mass m 5 5.00 kg attached to the end (Fig. P13.76). The object is placed at the level of the point of attachment with the spring unstretched, at position yi 5 L, and then it is released so that it swings like a pendulum. (a) Write Newtons second law symbolically for the system as the object passes through its lowest point. (Note that at the lowest point, r 5 L 2 yf .) (b) Write the conservation of energy equation symbolically, equating the total mechanical energies at the initial point and lowest point. (c) Find the coordinate position of the lowest point. (d) Will this pendulums period be greater or less than the period of a simple pendulum with the same mass m and length L? Explain.